Sizing and Painting a Rooftop Water Tank
A senior secondary school in Kanpur has to store water because the municipal supply arrives only on alternate days. A cylindrical tank of radius 1.4 m and height 2.5 m is to be placed on a square concrete platform measuring 3.5 m by 3.5 m and 0.2 m thick, already built on the roof. The school has 175 students on roll and expects each to use 4 litres a day; the kitchen and washing areas together take another 200 litres a day. The tank is empty at the start of a supply day and is filled completely. The plumber's quotation includes painting the outside curved surface and the flat top (not the base) at ₹120 per square metre, in a single coat. The principal also asks whether the circular base will sit safely inside the platform edge with a clear margin all round, and whether a second identical tank could be placed beside the first on the same platform if enrolment grows next year. Use π = 22/7.
1. Find the capacity of the tank in litres and the number of full days of supply it provides. Does it comfortably bridge the alternate-day supply?
Read the worked answer
Volume of a cylinder = πr²h. r² = 1.4 × 1.4 = 1.96 m²; then r²h = 1.96 × 2.5 = 4.9 m³. V = (22/7) × 4.9 = 22 × 0.7 = 15.4 m³.
Convert: 1 m³ = 1000 litres → 15.4 × 1000 = 15,400 litres.
Daily demand: Students: 175 × 4 = 700 L. Kitchen and washing: 200 L. Total = 900 L per day.
Days = 15,400 ÷ 900 = 17.11… After 17 full days, 17 × 900 = 15,300 L have been used, leaving 100 L - not enough for an eighteenth full day. So the tank serves 17 full days.
Judgement: the supply comes on alternate days, so the school needs only about 2 days of storage as a buffer. 17 days is far more than that, so the tank is oversized for daily use. It is, however, useful during exam weeks or power failures; if the school wanted only a 2-day buffer, a tank of about 1,800 L (roughly 1.8 m³) would do, and the money saved could go into a second filling point.
2. Compute the cost of painting the curved surface and the top, at ₹120 per square metre.
Read the worked answer
Curved surface area = 2πrh = 2 × (22/7) × 1.4 × 2.5. Work step by step: (22/7) × 1.4 = 4.4; 4.4 × 2.5 = 11; 11 × 2 = 22 m².
Top (one circle only) = πr² = (22/7) × 1.96 = 22 × 0.28 = 6.16 m².
The base rests on the platform, so it is not painted and is not counted. A very common error is to use the total surface area 2πr(r + h) = 2 × (22/7) × 1.4 × 3.9 ≈ 34.32 m², which paints a lid and a floor that do not exist.
Total painted area = 22 + 6.16 = 28.16 m². Cost = 28.16 × 120 = ₹3,379.20.
Judgement: since the quotation is for a single coat, ₹3,379.20 is the full figure. If two coats are needed the area is unchanged but the cost doubles to ₹6,758.40, and the school should confirm the number of coats in writing before work starts.
3. Will the tank sit safely on the platform with a margin all round? Could a second identical tank be placed on the same platform?
Read the worked answer
One tank: the base diameter is 2 × 1.4 = 2.8 m, against a platform side of 3.5 m. If the tank is centred, the margin is (3.5 − 2.8)/2 = 0.35 m all round. That is positive, so the base does fit - but 35 cm is a narrow walkway for maintenance, and the tank must be centred, not pushed to one edge.
Two tanks: this needs more than a quick comparison of 2.8 m with 3.5 m, because circles can be staggered. A circle of radius 1.4 m can be placed on a 3.5 m square only if its centre is at least 1.4 m from every edge. That restricts each centre to an inner square of side 3.5 − 2 × 1.4 = 0.7 m. Two centres inside a 0.7 m square can be at most 0.7√2 ≈ 0.99 m apart.
But two non-overlapping circles of radius 1.4 m need their centres at least 2.8 m apart.
0.99 m < 2.8 m, so two tanks cannot be placed on this platform however they are arranged. The platform would have to be at least about 5.6 m × 2.8 m. The principal should plan a second platform, not a second tank on the same one.